Question

Liquid water at \(25^{\circ} \mathrm{C}\) is introduced into an evacuated, insulated vessel. Identify the signs of the following thermodynamic functions for the process that occurs: \(\Delta H, \Delta S, \Delta T_{\text {water }}, \Delta S_{\text {surr }}, \Delta S_{\text {univ. }}\)

Short answer

The signs of the thermodynamic functions are as follows: \(\Delta H = 0\) (no heat exchange occurs); \(\Delta S > 0\) (entropy of the water increases due to expansion); \(\Delta T_{\text{water}} = 0\) (no heat transfer or work done); \(\Delta S_{\text{surr}} = 0\) (no heat exchange with surroundings); and \(\Delta S_{\text{univ}} > 0\) (overall entropy of the universe increases).

Step-by-step solution

Sign of \(\Delta H\)

Since the water is introduced into an evacuated, insulated vessel, there is no heat exchange with the surroundings. The process occurs at constant pressure, and since there is no heat exchange, the change in enthalpy (\(\Delta H\)), which is equal to the heat transferred at constant pressure, will be zero. In other words, \(\Delta H = 0\).

Sign of \(\Delta S\)

The entropy of the water (\(\Delta S\)) is a measure of its randomness during the process. Since the water is introduced into an evacuated vessel and allowed to expand (occupy more volume) at constant pressure, its randomness increases in the process. Therefore, the change in entropy of the water, \(\Delta S\), is positive.

Sign of \(\Delta T_{\text{water}}\)

The change in temperature of the water (\(\Delta T_{\text{water}}\)) depends on the heat exchange with the surroundings and the work done during the process. However, since the vessel is insulated, no heat transfer occurs. Additionally, since the water is expanding at constant pressure against the vacuum, no work is done (no external force against which work has to be done). As a result, the change in temperature \(\Delta T_{\text {water }}\) will be zero.

Sign of \(\Delta S_{\text {surr}}\)

For this process, the change in entropy of the surroundings (\(\Delta S_{\text{surr}}\)) is affected by the heat exchange with the water. As previously mentioned, the insulated vessel prevents any heat transfer between the water and the surroundings. With no heat exchange, there is no change in the entropy of the surroundings. Thus, \(\Delta S_{\text {surr }} = 0\).

Sign of \(\Delta S_{\text{univ}}\)

The change in entropy of the universe (\(\Delta S_{\text{univ}}\)) is the sum of the entropy changes for the water and the surroundings, i.e., \(\Delta S_{\text{univ}} = \Delta S_{\text{water}} + \Delta S_{\text{surr}}\). Since the change in entropy for the water is positive and the change in entropy for the surroundings is zero, the overall change in entropy of the universe will be positive. Thus, \(\Delta S_{\text {univ }} > 0\).

Key concepts

Entropy Change

When we talk about entropy change in thermodynamics, we're referring to the measure of disorder or randomness within a system. Entropy is fundamentally connected to the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time.

In the example provided, the introduction of liquid water into an evacuated, insulated vessel leads to an increase in entropy, signified by a positive \(\Delta S\). This occurs because when the water expands, its molecules spread out and occupy more volume, resulting in increased random motion and thus, a higher degree of disorder. In a sense, entropy can be thought of as the natural tendency of things to spread out and disperse.

Enthalpy Change

Enthalpy change, represented by \(\Delta H\), is a key concept in thermodynamics that reflects the total heat content of a system. It's indicative of both internal energy of the system and the product of pressure and volume. Crucially, \(\Delta H\) also represents the heat absorbed or released by the system at constant pressure.

In our specific scenario, an insulated vessel ensures no heat transfer occurs with the surroundings, implying no change in enthalpy—thus, \(\Delta H = 0\). This concept emphasizes the closed nature of the system and the conservation of energy within it. Despite changes internally, the overall energy in the form of enthalpy remains constant because the system is not interacting with its external environment in terms of heat transfer.

Isolated System

An isolated system in thermodynamics is a system that does not exchange matter or energy with its surroundings. In the example of the liquid water, the vessel is not only evacuated, meaning no matter can enter or leave, but it is also insulated, meaning no heat can enter or leave.

This fits the definition of an isolated system, where all exchanges are internal, and the system's total mass and energy remain constant. These conditions make analyzing thermodynamic problems simpler because they remove the complexities of interaction with the environment. It is essential to note, however, that perfect isolated systems do not exist in reality—they are an idealization used for theoretical purposes.

Heat Transfer

Heat transfer is a fundamental process in thermodynamics where thermal energy moves from a hotter region to a colder one, seeking equilibrium. This transfer can occur in various ways, including conduction, convection, and radiation.

In the mentioned exercise, the context is an insulated vessel, which impedes the flow of heat to or from the system, thereby preventing any heat transfer. Understanding heat transfer is crucial in many applications, including designing efficient cooling systems, insulating buildings, and studying climate change. The absence of heat transfer in such an insulated system underscores the significance of boundary conditions in thermodynamic analyses, as they dictate how a system will behave and interact—or not interact—with its environment.